Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More one can be obtained from the other by uniformly scaling with additional translation and reflection; this means that either object can be rescaled and reflected, so as to coincide with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level. For example, all circles are similar to each other, all squares are similar to each other, all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, isosceles triangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle the triangles are similar.
Corresponding sides of similar polygons are in proportion, corresponding angles of similar polygons have the same measure. This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are similar, but some school textbooks exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar. In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional, it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several statements each of, necessary and sufficient for two triangles to be similar: The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:If ∠BAC is equal in measure to ∠B′A′C′, ∠ABC is equal in measure to ∠A′B′C′ this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar. All the corresponding sides have lengths in the same ratio:AB/A′B′ = BC/B′C′ = AC/A′C′; this is equivalent to saying. Two sides have lengths in the same ratio, the angles included between these sides have the same measure. For instance:AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′; this is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; when two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′. There are several elementary results concerning similar triangles in Euclidean geometry: Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other.
Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if one other side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, find a point F such that △ABC ∼ △DEF; the statement that the point F satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by G. D. Birkhoff the SAS similarity criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms. Similar triangles provide the basis for many synthetic proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles provide the foundations for right triangle trigonometry.
The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles. Equality of all angles in sequence is not sufficient to guarantee similarity. A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional. For given n, all regular n-gons are similar. Several types of curves have the property; these include: Circles Parabolas Hyperbolas of a specific eccentricity Ellipses of a specific eccentricity Catenaries Graphs of the logarithm function for different bases Graphs of the exponential function for different bases Logarithmic spirals are self-similar A similarity of a Euclidean space is a bijection f from the space onto itself that multiplies all distances
A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave, it is named after the function sine. It occurs in pure and applied mathematics, as well as physics, signal processing and many other fields, its most basic form as a function of time is: y = A sin = A sin where: A, the peak deviation of the function from zero. F, ordinary frequency, the number of oscillations that occur each second of time. Ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second φ, specifies where in its cycle the oscillation is at t = 0. When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ /ω seconds. A negative value represents a delay, a positive value represents an advance; the sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform; this property makes it acoustically unique.
In general, the function may have: a spatial variable x that represents the position on the dimension on which the wave propagates, a characteristic parameter k called wave number, which represents the proportionality between the angular frequency ω and the linear speed ν. The wavenumber is related to the angular frequency by:. K = ω v = 2 π f v = 2 π λ where λ is the wavelength, f is the frequency, v is the linear speed; this equation gives a sine wave for a single dimension. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wavenumber k are interpreted as vectors, their product as a dot product. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed; this wave pattern occurs in nature, including wind waves, sound waves, light waves. A cosine wave is said to be sinusoidal, because cos = sin , a sine wave with a phase-shift of π/2 radians.
Because of this head start, it is said that the cosine function leads the sine function or the sine lags the cosine. The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics. To the human ear, a sound, made of more than one sine wave will have perceptible harmonics. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, the reason why the same musical note played on different instruments sounds different. On the other hand, if the sound contains aperiodic waves along with sine waves the sound will be perceived to be noisy, as noise is characterized as being aperiodic or having a non-repetitive pattern. In 1822, French mathematician Joseph Fourier discovered that sinusoidal waves can be used as simple building blocks to describe and approximate any periodic waveform, including square waves. Fourier used it as an analytical tool in the study of waves and heat flow, it is used in signal processing and the statistical analysis of time series.
Since sine waves propagate without changing form in distributed linear systems, they are used to analyze wave propagation. Sine waves traveling in two directions in space can be represented as u = A sin When two waves having the same amplitude and frequency, traveling in opposite directions, superpose each other a standing wave pattern is created. Note that, on a plucked string, the interfering waves are the waves reflected from the fixed end
In mathematics, a Lissajous curve known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equations x = A sin , y = B sin , which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, in more detail by Jules Antoine Lissajous in 1857; the appearance of the figure is sensitive to the ratio a/b. For a ratio of 1, the figure is an ellipse, with special cases including lines. Another simple Lissajous figure is the parabola. Other ratios produce more complicated curves; the visual form of these curves is suggestive of a three-dimensional knot, indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures. Visually, the ratio a/b determines the number of "lobes" of the figure. For example, a ratio of 3/1 or 1/3 produces a figure with three major lobes. A ratio of 5/4 produces a figure with five horizontal lobes and four vertical lobes. Rational ratios produce closed or "still" figures, while irrational ratios produce figures that appear to rotate.
The ratio A/B determines the relative width-to-height ratio of the curve. For example, a ratio of 2/1 produces a figure, twice as wide as it is high; the value of δ determines the apparent "rotation" angle of the figure, viewed as if it were a three-dimensional curve. For example, δ = 0 produces x and y components that are in phase, so the resulting figure appears as an apparent three-dimensional figure viewed from straight on. In contrast, any non-zero δ produces a figure that appears to be rotated, either as a left–right or an up–down rotation. Lissajous figures where a = 1, b = N and δ = N − 1 N π 2 are Chebyshev polynomials of the first kind of degree N; this property is exploited to produce a set of points, called Padua points, at which a function may be sampled in order to compute either a bivariate interpolation or quadrature of the function over the domain ×. The relation of some Lissajous curves to Chebyshev polynomials is clearer to understand if the Lissajous curve which generates each of them is expressed using cosine functions rather than sine functions.
X = cos , y = cos The animation shows the curve adaptation with continuously increasing a/b fraction from 0 to 1 in steps of 0.01. Below are examples of Lissajous figures with δ = π/2, an odd natural number a, an natural number b, |a − b| = 1. Prior to modern electronic equipment, Lissajous curves could be generated mechanically by means of a harmonograph. Lissajous curves can be generated using an oscilloscope. An octopus circuit can be used to demonstrate the waveform images on an oscilloscope. Two phase-shifted sinusoid inputs are applied to the oscilloscope in X-Y mode and the phase relationship between the signals is presented as a Lissajous figure. In the professional audio world, this method is used for realtime analysis of the phase relationship between the left and right channels of a stereo audio signal. On larger, more sophisticated audio mixing consoles an oscilloscope may be built-in for this purpose. On an oscilloscope, we suppose x is CH1 and y is CH2, A is the amplitude of CH1 and B is the amplitude of CH2, a is the frequency of CH1 and b is the frequency of CH2, so a/b is the ratio of frequencies of the two channels, δ is the phase shift of CH1.
A purely mechanical application of a Lissajous curve with a = 1, b = 2 is in the driving mechanism of the Mars Light type of oscillating beam lamps popular with railroads in the mid-1900s. The beam in some versions traces out a lopsided figure-8 pattern on its side; when the input to an LTI system is sinusoidal, the output is sinusoidal with the same frequency, but it may have a different amplitude and some phase shift. Using an oscilloscope that can plot one signal against another to plot the output of an LTI system against the input to the LTI system produces an ellipse, a Lissajous figure for the special case of a = b; the aspect ratio of the resulting ellipse is a function of the phase shift between the input and output, with an aspect ratio of 1 corresponding to a phase shift of ±90° and an aspect ratio of ∞ corresponding to a phase shift of 0° or 180°. The figure below summarizes; the phase shifts are all negative so. The arrows show the direction of rotation of the Lissajous figure. A Lissajous curve is used in experimental tests to determine if a device may be properly categorized as a memristor.
Lissajous figures were sometimes displayed on oscilloscopes meant to simulate high-tech equipment in science-fiction TV shows and movies in the 1960s and 1970s. The title sequence by John Whitney for Alfred Hitchcock's 1958 film Vertigo is based on Lissajous figures. In a sequence towards the end of an episode of Columbo entitled "Make me a Perfect Murder", the
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Luigi Guido Grandi
Dom Guido Grandi, O. S. B. Cam. was an Italian monk, philosopher, theologian and engineer. Grandi was born on 1 October 1671 in Cremona and christened Luigi Francesco Lodovico; when he was of age, he was educated at the Jesuit college there. After he completed his studies there in 1687, he entered the novitiate of the Camaldolese monks at Ferrara and took the name of Guido. In 1693 he was sent to the Monastery of St. Gregory the Great, the Camaldolese house in Rome, to complete his studies in philosophy and theology in preparation for Holy Orders. A year Grandi was assigned as professor of both fields at the Camaldolese Monastery of St. Mary of the Angels in Florence, it appears. He did his research however, as he was appointed professor of philosophy at St. Gregory Monastery in 1700, subsequently holding a post in the same field in Pisa. By 1707, Dom Grandi had developed such a reputation in the field of mathematics that he was named court mathematician to the Grand Duke of Tuscany, Cosimo III de Medici.
In that post, he worked as an engineer, being appointed Superintendent of Water for the Duchy, in that capacity he was involved in the drainage of the Chiana Valley. In 1709 he visited England where he impressed his colleagues there, as he was elected a Fellow of the Royal Society; the University of Pisa named him Professor of Mathematics in 1714. It was there that he died on 4 July 1742. In 1701 Grandi published a study of the conical loxodrome, followed by a study in 1703 of the curve which he named versiera, from the Latin: vertere; this curve was studied by one of the few women scientists to achieve a degree, Maria Gaetana Agnesi. Through a mistranslation by the translator of her work into English who mistook the term "witch" for Grandi's term, this curve became known in English as the witch of Agnesi, it was through his studies on this curve that Grandi helped introduce Leibniz' ideas on calculus to Italy. In mathematics Grandi is best known for his work Flores geometrici, studying the rose curve, a curve which has the shape of a petalled flower, for Grandi's series.
He named. He contributed to the Note on the Treatise of Galileo Concerning Natural Motion in the first Florentine edition of Galileo Galilei's works. Geometrica demonstratio Vivianeorum problematum. Florentiae: ex Typographia Iacobi de Guiduccis propè Conductam. 1699. De infinitis infinite parvorum ordinibus disquisitio geometrica. Pisis: ex Typographia Francisci Bindi impress. Archiepisch. 1710. Epistola mathematica de momento gravium in planis inclinatis. Lucae: typis Peregrini Frediani. 1711. Dialoghi circa la controversia eccitatagli contro dal sig. Alessandro Marchetti. In Lucca: ad istanza di Francesco Maria Gaddi librajo in Pisa. 1712. Prostasis ad exceptiones clari Varignonii libro De infinitis infinitorum ordinibus oppositas circa magnitudinum plusquam-infinitarum Vallisii defensionem et anguli contactus. Pisis: ex Typographia Francisci Bindi impress. Archiepisch. 1713. Del movimento dell'acque trattato geometrico. Firenze. Relazione delle operazioni fatte circa il padule di Fucecchio. In Lucca: per Leonardo Venturini.
1718. Trattato delle resistenze. Firenze: per Tartini e Franchi. 1718. Compendio delle Sezioni coniche d'Apollonio con aggiunta di nuove proprietà delle medesime sezioni. In Firenze: nella Stamperia di S. A. R. per gli Tartini e Franchi. 1722. Instituzioni meccaniche. In Firenze: nella Stamperia di S. A. R. per Gio: Gaetano Tartini e Santi Franchi. 1739. Istituzioni di aritmetica pratica. In Firenze: nella Stamperia di S. A. R. per Gio: Gaetano Tartini e Santi Franchi. 1740. Sectionum conicarum synopsis. Florentiae: ex typographio Ioannis Paulli Giovannelli. 1750. O'Connor, John J.. Galileo Project: Guido Grandi
In mathematics, a rose is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are related to free groups. A rose; that is, the rose is the quotient space C/S, where C is a disjoint union of circles and S a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, one edge for each circle; this makes it a simple example of a topological graph. A rose with n petals can be obtained by identifying n points on a single circle; the rose. The fundamental group of a rose is free, with one generator for each petal; the universal cover is an infinite tree, which can be identified with the Cayley graph of the free group. The intermediate covers of the rose correspond to subgroups of the free group; the observation that any cover of a rose is a graph provides a simple proof that every subgroup of a free group is free Because the universal cover of a rose is contractible, the rose is an Eilenberg–MacLane space for the associated free group F.
This implies that the cohomology groups Hn are trivial for n ≥ 2. Any connected graph is homotopy equivalent to a rose; the rose is the quotient space of the graph obtained by collapsing a spanning tree. A disc with n points removed deformation retracts onto a rose with n petals. One petal of the rose surrounds each of the removed points. A torus with one point removed deformation retracts onto a figure eight, namely the union of two generating circles. More a surface of genus g with one point removed deformation retracts onto a rose with 2g petals, namely the boundary of a fundamental polygon. A rose can have infinitely many petals, leading to a fundamental group, free on infinitely many generators; the rose with countably infinitely many petals is similar to the Hawaiian earring: there is a continuous bijection from this rose onto the Hawaiian earring, but the two are not homeomorphic. Bouquet graph Quadrifolium Free group Topological graph Hatcher, Algebraic topology, Cambridge, UK: Cambridge University Press, ISBN 0-521-79540-0 Munkres, James R. Topology, Englewood Cliffs, N.
J: Prentice Hall, Inc, ISBN 0-13-181629-2 Stillwell, Classical topology and combinatorial group theory, Berlin: Springer-Verlag, ISBN 0-387-97970-0